#2255 ⟨a, b | ab=aa, bba=bb⟩

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Contents

  1. Properties
  2. Finite structure
  3. Cayley table
  4. Right Cayley graph
  5. Left Cayley graph
  6. Rewriting system
  7. Same cardinality

Properties

Finite structure

Elements in the center commute with all other elements.
A left zero element x satisfies xy = x for all y.
An idempotent element x satisfies x2 = x.
The index and period of x is the least m (index) and n (period) such that x(m+n) = xm.

Cayley table

Idempotents are shown in bold.

1ababbab2ab2babbab2
11ababbab2ab2babbab2
aaababab2ab2ab2ab2ab2ab2
bbbab2babb2b2bab2b2b2
ababab2ab2ab2ab2ab2ab2ab2ab2
babababbabbab2bab2bab2bab2bab2bab2
b2b2b2b2b2b2b2b2b2b2
ab2ab2ab2ab2ab2ab2ab2ab2ab2ab2
babbabbab2bab2bab2bab2bab2bab2bab2bab2
bab2bab2bab2bab2bab2bab2bab2bab2bab2bab2

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Rewriting system

Format:
Word:
Enter a word above to compute its normal form. Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Strategy:
Result: 1 
1
#RuleProof
1. a2 ⇒ ab [1]
2. b3 ⇒ b2 [4]
3. b2a ⇒ b2 [2]
4. aba ⇒ ab2 [3]
# ab:ab=aa,bba=bb ba - -
aa=ab
bbb=bb
bba=bb
aba=abb

Same cardinality

26 unique, 942 total

Σ#PresentationPropertiesDescriptionφ
7118⟨a, b | aaa=b, bbb=1⟩Grp Com 9ℤ9562
8568⟨a, b | aaa=a, abb=b⟩Noc 95
8571⟨a, b | aaa=a, bbb=a⟩Noc Com 9ℕ(9 = 3)48
8581⟨a, b | aaa=b, bbb=a⟩Noc Com 9ℕ(9 = 1)62
8601⟨a, b | aba=b, bab=a⟩Noc 93
91584⟨a, b | aaa=aa, bbb=a⟩Noc Com 9ℕ(9 = 6)16
91595⟨a, b | aaa=ab, bab=b⟩Noc 99
91598⟨a, b | aaa=ab, bbb=a⟩Noc Com 9ℕ(9 = 4)40
91608⟨a, b | aaa=bb, bbb=a⟩Noc Com 9ℕ(9 = 2)70
91991⟨a, b | aaa=a, abba=b⟩Noc 93
92064⟨a, b | aab=b, abba=a⟩Noc 938
104125⟨a, b | aab=aaa, baa=b⟩Noc 92
104130⟨a, b | aab=aaa, bbb=a⟩Noc Com 9ℕ(9 = 7)1
104156⟨a, b | abb=aaa, bbb=a⟩Noc Com 9ℕ(9 = 5)12
105035⟨a, b | aaa=aa, abbb=b⟩Noc 93
105079⟨a, b | aaa=bb, aaba=b⟩Noc Com 98
105124⟨a, b | aab=aa, bbbb=a⟩Noc Com 9ℕ(9 = 8)5
105339⟨a, b | aaa=bb, aab=ab⟩Noc 91
106824⟨a, b | aab=b, bbb=aaa⟩Noc Com 95
107139⟨a, b | bb=aa, aaaa=ab⟩Noc 93
109327⟨a, b | ab=a, baaa=bbb⟩Noc 93
109359⟨a, b | ab=a, bbaa=bbb⟩Noc 92
1119506⟨a, b | aab=b, aaaaa=ba⟩Noc 9
1119507⟨a, b | aab=b, aaaaa=bb⟩Noc Com 9
1120723⟨a, b | ab=aa, aaaaa=bb⟩Noc 915
1125656⟨a, b | ab=a, bbbbb=baa⟩Noc 9